论文标题
某些Smash产品的Grothendieck代数半烟霍普夫代数
The Grothendieck algebras of certain smash product semisimple Hopf algebras
论文作者
论文摘要
让$ h $是特征性$ p> p> \ dim _ {\ mathbbm {k}}（h）^{1/2} $ p \ nmid 2 \ nmid 2 \ dim _ _ {kmmathbm {kmmmathbm {km {在本文中，我们考虑了Smash Product Semimple Hopf代数$ H \＃\ MathBbm {K} G $，其中$ G $是$ n：= 2 \ dim _ {\ MathBbm {k}}（k}}}（h）$的订单$ n：= 2 \ dim _ {\ dim _ {\ dim _ {使用$ h $的不可约表示和$ \ mathbbm {k} g $的表示，我们确定$ h \＃\＃\ mathbbm {K k} g $的所有非同构不可修复表示。 Grothendieck algebra $（g_0（h \＃\ Mathbbm {k} g） $（g_0（h）\ otimes _ {\ mathbb {z}}} \ Mathbbm {k}，*）$ h $。要建立此连接，我们在$ g_0（h）\ otimes _ {\ mathbb {z}} \ mathbbm {k} $上赋予了新的乘法运算符$ \ star $ $（g_0（h \＃\ mathbbm {k} g）\ otimes _ {\ mathbb {z}}} \ mathbbm {k}，\ ast）$是与直接总和$（g_0（h）\ otimes _ {\ mathbb {z}}} \ mathbbm {k}，*）^{\ oplus \ frac {n} {2}}}} $ and $（g_0（h）\ otimes _ {\ mathbb {z}}} \ mathbbm {k}，\ star）^{\ oplus \ frac {n} {2}}} $。
Let $H$ be a semisimple Hopf algebra over an algebraically closed field $\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$ and $p\nmid 2\dim_{\mathbbm{k}}(H)$. In this paper, we consider the smash product semisimple Hopf algebra $H\#\mathbbm{k}G$, where $G$ is a cyclic group of order $n:=2\dim_{\mathbbm{k}}(H)$. Using irreducible representations of $H$ and those of $\mathbbm{k}G$, we determine all non-isomorphic irreducible representations of $H\#\mathbbm{k}G$. There is a close relationship between the Grothendieck algebra $(G_0(H\#\mathbbm{k}G)\otimes_{\mathbb{Z}}\mathbbm{k},*)$ of $H\#\mathbbm{k}G$ and the Grothendieck algebra $(G_0(H)\otimes_{\mathbb{Z}}\mathbbm{k},*)$ of $H$. To establish this connection, we endow with a new multiplication operator $\star$ on $G_0(H)\otimes_{\mathbb{Z}}\mathbbm{k}$ and show that the Grothendieck algebra $(G_0(H\#\mathbbm{k}G)\otimes_{\mathbb{Z}}\mathbbm{k},\ast)$ is isomorphic to the direct sum of $(G_0(H)\otimes_{\mathbb{Z}}\mathbbm{k},*)^{\oplus\frac{n}{2}}$ and $(G_0(H)\otimes_{\mathbb{Z}}\mathbbm{k},\star)^{\oplus\frac{n}{2}}$.
